Difference equation for the Heckman–Opdam hypergeometric function and its confluent Whittaker limit

A Connection Formula for the q-Confluent Hypergeometric Function

We show a connection formula for the q-confluent hypergeometric functions 2φ1(a, b; 0; q, x). Combining our connection formula with Zhang’s connection formula for 2φ0(a, b;−; q, x), we obtain the connection formula for the q-confluent hypergeometric equation in the matrix form. Also we obtain the connection formula of Kummer’s confluent hypergeometric functions by taking the limit q → 1− of our...

On the confluent hypergeometric function coming from the Pareto distribution

Making use of the confluent hypergeometric function we can obtain the Laplace-Stieltje transform of the Pareto distribution in the following form ζ(s) = hU(1; 1− h; s) = 1F1(1; 1− h; s)− Γ(1− h)s1F1(1 + h; 1 + h; s). About this transform, we obtain an identity, Γ(1 + h)|U(1, 1− h, s)|2 = ∫ ∞ 0 ∫ ∞ 0 λhe−λ−y |λ+ s|2 + λy 2000 Mathematical Subject Classification: 33C15, 60E07

Properties of the Bivariate Confluent Hypergeometric Function Kind 1 Distribution

The bivariate confluent hypergeometric function kind 1 distribution is defined by the probability density function proportional to x1 1 x2 2 1 F1(α; β; −x1 − x2). In this article, we study several properties of this distribution and derive density functions of X1/X2, X1/(X1 + X2), X1 + X2 and 2 √ X1X2. The density function of 2 √ X1X2 is represented in terms of modified Bessel function of the s...

Multivariate Generalization of the Confluent Hypergeometric Function Kind 1 Distribution

The confluent hypergeometric function kind 1 distribution with the probability density function pdf proportional to x −11F1 α; β;−x , x > 0 occurs as the distribution of the ratio of independent gamma and beta variables. In this article, a multivariate generalization of this distribution is defined and derived. Several pertinent properties of this multivariate distribution are discussed that sh...

Gauss Quadrature Approximations to Hypergeometric and Confluent Hypergeometric Functions